Homogeneous partial differential equations for superpositions of indeterminate functions of several variables

We determine essentially all partial differential equations satisfied by superpositions of tree type and of a further special type. These equations represent necessary and sufficient conditions for an analytic function to be locally expressible as an analytic superposition of the type indicated. The representability of a real analytic function by a superposition of this type is independent of whether that superposition involves real-analytic functions or $C^{\rho}$-functions, where the constant $\rho$ is determined by the structure of the superposition. We also prove that the function $u$ defined by $u^n=xu^a+yu^b+zu^c+1$ is generally non-representable in any real (resp. complex) domain as $f\bigl(g(x,y),h(y,z)\bigr)$ with twice differentiable $f$ and differentiable $g$, $h$ (resp. analytic $f$, $g$, $h$).